DSpace at VNU: Functional inequalities and theorems of the alternative involving composite functions

DSpace at VNU: Functional inequalities and theorems of the alternative involving composite functions tài liệu, giáo án,...

J Glob Optim DOI 10.1007/s10898-013-0100-z Functional inequalities and theorems of the alternative involving composite functions N Dinh · G Vallet · M Volle Received: 30 June 2012 / Accepted: August 2013 © Springer Science+Business Media New York 2013 Abstract We propose variants of non-asymptotic dual transcriptions for the functional inequality of the form f + g + k ◦ H ≥ h The main tool we used consists in purely algebraic formulas on the epigraph of the Legendre-Fenchel transform of the function f + g + k ◦ H that are satisfied in various favorable circumstances The results are then applied to the contexts of alternative type theorems involving composite and DC functions The results cover several Farkas-type results for convex or DC systems and are general enough to face with unpublished situations As applications of these results, nonconvex optimization problems with composite functions, convex composite problems with conic constraints are examined at the end of the paper There, strong duality, stable strong duality results for these classes of problems are established Farkas-type results and stable form of these results for the corresponding systems involving composite functions are derived as well Keywords Functional inequalities · Alternative type theorems · Farkas-type results · Stable strong duality · Stable Farkas lemma · Set containments · Nonconvex composite optimization problems · Convex composite problems with conic constraints Mathematics Subject Classification 39B62 · 49J52 · 46N10 N Dinh (B) Department of Mathematics, International University, VNU-HCM, Thuduc district, Ho Chi Minh City, Vietnam e-mail: ndinh@hcmiu.edu.vn G Vallet Laboratory of Applied Mathematics, UMR-CNRS 5142, University of Pau, IPRA, BP 1155, 64013 Pau Cedex, France e-mail: guy.vallet@univ-pau.fr M Volle Laboratoire de Mathématiques d’Avignon, Université d’Avignon, 33 rue Pasteur, 84000 Avignon, France e-mail: michel.volle@univ-avignon.fr 123 J Glob Optim Introduction Given two proper convex functions a, b : X → R ∪ {+∞} on a locally convex Hausdorff topological vector space (l.c.H.t.v.s.) X , the relation (a + b)∗ (x ∗ ) = a ∗ (x ∗ − y ∗ ) + b∗ (y ∗ ) ∗ ∗ y ∈X (1) has been used in [10] for the usual formula on the -subdifferential of the sum of the proper convex function a + b It should be emphasized that a classical sufficient condition for the validity of (1) is that a is finite and continuous at a point of dom b (see [24, Theorem 2.8.7]) It is easy to see that (1) is equivalent to epi (a + b)∗ = epi a ∗ + epi b∗ (2) In the case when a, b ∈ Γ (X ) and a + b is proper, it was observed in [3, Theorem 2.1] that epi (a + b)∗ = w ∗ − cl(epi a ∗ + epi b∗ ) and, consequently, that (2) is equivalent to epi a ∗ + epi b∗ is w ∗ -closed A generalization of the previous result has been established in [2, Theorem 9.2] which n dom a = ∅, and Δ a nonempty reads as follows: Given a1 , , an ∈ Γ (X ) such that ∩i=1 i ∗ subset of X , the following statements are equivalent: (i) (ii) ∗ n n ∗ ∗ = i=1 i=1 (x i ), n ∗ ∗ epi a in w -closed regarding i=1 i n ∗ i=1 x i = x ∗ , ∀x ∗ ∈ Δ, the set Δ × R Let us quote that condition (i) above amounts to saying that ∗ n (Δ × R) ∩ epi i=1 n = (Δ × R) ∩ epi ai∗ (3) i=1 The present paper is devoted to characterizations of relations like (2) or (3) in various situations involving composite functions and in the absence of the convexity and lower semi-continuity of the functions involved More precisely, we start by establishing dual transcriptions for the functional inequality of the form (I) f (x) + g(x) + (k ◦ H )(x) ≥ h(x), ∀x ∈ X where f, g, h : X → R ∪ {+∞} are extended real-valued functions defined on the l.c.H.t.v.s X, H : dom H ⊂ X → Z is a mapping defined on a non-void subset dom H of X , with values in another l.c.H.t.v.s Z , and k : Z → R ∪ {+∞} is an extended real-valued function One may consider the mapping H defined on the whole space X with values on Z • := Z ∪{∞ Z } and H (x) = ∞ Z when x ∈ dom H , where the element ∞ Z does not belong to Z We then show that relations like (2) or (3) involving composite functions are necessary and sufficient conditions for such transcriptions It is worth mentioning that several of these characterizations hold without any convexity and lower semi-continuity assumptions The reason why we study relations like (I ) is that this frame encompasses most of generalized versions of the Farkas Lemma (see, e.g., [2,8,11]), together with convex/concave containments (see, e.g., [12,17]) In fact, the relation (I ) is sufficiently general to face also with unpublished circumstances (see Sect 6) 123 J Glob Optim Fig Double composite model The recent papers [4,6] partially devoted to asymptotic dual transcriptions of such a functional inequality with or without convexity, lower semicontinuity assumptions on the functions involved, and without any constraint qualification conditions There, the results were then applied to derive sequential Lagrange multipliers, duality results for general optimization problems without convexity nor constraint qualification conditions, asymptotic Farkas lemmas for nonconvex systems, for linear infinite systems without qualification conditions The abilities of applying these results to other optimization problems such as variational inequalities, equilibrium problems were shown in [4,6] as well The aim of the present paper is twofold We first provide purely algebraic necessary and sufficient conditions for the validity of non-asymptotic dual transcriptions (Theorems 2–3) The counter part of such results are obtained in the presence of convexity and lower semicontinuity was given in Sect which gives rise to variants of characterizations of Farkastype results (Theorem 5, Corollary 3) and strong duality and stable strong duality for convex and DC optimization problems and also stable Farkas-type results for systems involving convex systems or systems with DC functions (Corollaries 4–5) The second purpose of the paper is to apply our approach to the context of the alternative theorems (Theorems 10–14, Corollaries 6–7, Theorem 15), to convex/concave containments (Proposition 2) These are given in Sect To show the possibilities of using the results obtained to get various new results and applications, we consider the use of “double composite model” of the form f +g+ p◦ M +q ◦ N ≥h (4) corresponding to the diagram shown in Fig Where U, V are l.c.H.t.v.s with U • := U ∪ {∞U }, V • := V ∪ {∞V }, and ∞U , ∞V are elements that not belong to U and V (resp.) while U+ and V+ are closed convex cones in U and V , respectively If we set H := (M, N ) and k(u, v) = p(u) + q(v) then (4) is nothing but (I ) This will be realized in Sect There, some special cases will be examined to get strong duality, stable strong duality for nonconvex and convex optimization problems involving composite functions New versions of Farkas-type results and their stable versions are proposed as well Notations and preliminary results Throughout this paper f, g, h : X → R ∪ {+∞} are extended real-valued functions defined on the locally convex Hausdorff topological vector space X with its topological dual X ∗ 123 J Glob Optim equipped with the weak∗ -topology, H : dom H ⊂ X → Z is a mapping defined on a nonvoid subset dom H of X , with values in an other locally convex Hausdorff topological vector space Z , and k : Z → R ∪ {+∞} is an extended real-valued function Let us now introduce some usual notations For a subset B of X , the indicator function of B, denoted by i B , is defined by i B (x) = if x ∈ B and i B (x) = +∞ if x ∈ B For any function ϕ : X → R ∪ {+∞}, the epigraph of ϕ is defined as epi ϕ := {(w, r ) ∈ X × R | ϕ(w) ≤ r } while the strict epigraph of ϕ is epi s ϕ := {(w, r ) ∈ X × R | ϕ(w) < r } Epigraphs of extended real-valued functions defined on the dual space X ∗ will be defined in the same way Let us recall that the Legendre-Fenchel transform of a given a : X → R ∪ {+∞} is defined as (see, e.g., [24]) a ∗ (w) = sup ( w, x − a(x)) , ∀w ∈ X ∗ x∈X while the conjugate of any ϕ : X ∗ → R is given by ϕ ∗ (x) = sup ( w, x − ϕ(w)) , ∀x ∈ X w∈X ∗ It is worth noticing that if k is a proper function on X, k ∗ does not take the value −∞ and so, for any λ ∈ dom k ∗ , k ∗ (λ) ∈ R Moreover, if k is a proper lower semi-continuous (lsc) convex function, in other words k belongs to the set Γ (X ) of extended real-valued proper, lower semi-continuous convex functions on X , then k coincides with its Legendre-Fenchel biconjugate, i.e., k = k ∗∗ The infimal convolution ϕ ✷ψ of the functions ϕ, ψ : X ∗ → R is given by (ϕ ✷ψ)(w) := inf ∗ (ϕ(w − v) + ψ(v)) , ∀w ∈ X ∗ v∈X We will use the classical properties below: for any a, b : X −→ R ∪ {+∞} with dom a = ∅ and dom b = ∅, it holds epi a ∗ + epi b∗ ⊂ epi (a ∗ ✷b∗ ) ⊂ epi (a ∗∗ + b∗∗ )∗ ⊂ epi (a + b)∗ Let S be a convex cone in Z , i.e., S is a convex set and (α, u) ∈ [0, +∞[×S ⇒ αu ∈ S Then Z will be preordered by the reflexive and transitive relation: u ≤ S z ⇔ z − u ∈ S, ∀u, z ∈ Z The positive polar cone of S, denoted by S ∗ , is defined as S ∗ := {z ∗ ∈ Z ∗ | z ∗ , s ≥ 0, ∀s ∈ S} Now let H : dom H ⊂ X → Z The S-epigraph of H is defined by (see [16]) epi S H = {(x, z) ∈ dom H × Z | z ∈ H (x) + S} 123 (5) J Glob Optim If H has closed S-epigraph then it is called S-epi-closed The mapping H is said to be S-convex if dom H is convex and H (tu + (1 − t)z) ≤ S t H (u) + (1 − t)H (z), ∀u, z ∈ dom H, ∀t ∈ [0, 1] Moreover, a function k : Z → R ∪ {+∞} is called increasing with respect to S (or, briefly S-increasing) if u, z ∈ dom k, u ≤ S z implies k(u) ≤ k(z) For a mapping H : dom H ⊂ X → Z and a function k : Z → R ∪ {+∞}, let us set k(H (x)) if x ∈ dom H +∞ otherwise, (k ◦ H )(x) = and for the sake of convenience, we write λH instead of λ ◦ H for any λ ∈ Z ∗ , λ, H (x) +∞ (λH )(x) := if x ∈ dom H, else, where Z ∗ is the topological dual of the space Z The following classical theorem will be used in the sequel Theorem [24, Theorem 2.8.10] Let a : X −→ R ∪ {+∞}, k : Z −→ R ∪ {+∞} be proper convex functions and let H : dom H ⊂ X → Z be S-convex Assume that k is finite and continuous at a point of H (dom H ∩ dom a) and k is increasing with respect to S on H (dom H ∩ dom a) + S We then have, for all w ∈ X ∗ , (a + k ◦ H )∗ (w) = min∗ [k ∗ (λ) + (a + λH )∗ (w)], λ∈S or, equivalently epi (a + k ◦ H )∗ = epi [a + λH − k ∗ (λ)]∗ λ∈S ∗ ∩dom k ∗ For the sake of convenience, the following conventions will be used throughout the paper: • ∪i∈I Ai = ∅ whenever I = ∅, • for any E ⊂ X ∗ × R, ∅ + E = E + ∅ = ∅, • (+∞) + (−∞) = (+∞) + (+∞) = +∞ The use of the Legendre-Fenchel transform in a purely algebraic setting Although the functions f, g, λH (λ ∈ Z ∗ ) are not necessarily convex, the use of their conjugates remain relevant, as shown by the next lemma Lemma One always has A := epi f ∗ + epi g ∗ + epi λH − k ∗ (λ) ∗ λ∈dom k ∗ ⊂ B := epi f ∗ + epi g + λH − k ∗ (λ) ∗ λ∈dom k ∗ ∗ ⊂ epi ( f + g + k ◦ H ) So, if A = epi ( f + g + k ◦ H )∗ then ∗ ( f + g + k ◦ H )∗ = f ∗∗ + g ∗∗ + k ∗∗ ◦ H 123 J Glob Optim Proof • A ⊂ B One has epi g ∗ + epi λH − k ∗ (λ) ∗ epi g ∗ + epi λH − k ∗ (λ) = λ∈dom k ∗ ∗ λ∈dom k ∗ and, by (5) with a = g and b = λH − k ∗ (λ) A ⊂ epi f ∗ + for any λ ∈ dom k ∗ , epi g + λH − k ∗ (λ) ∗ λ∈dom k ∗ • B ⊂ epi ( f + g + k ◦ H )∗ We first observe that for any λ ∈ dom k ∗ and x ∈ dom H , one has −k ∗ (λ) ≤ (k ◦ H )(x) − (λH )(x) and so, λH − k ∗ (λ) ≤ k ◦ H on dom H We thus have [g + λH − k ∗ (λ)]∗ ≥ (g + k ◦ H )∗ , ∀λ ∈ dom k ∗ , and therefore, epi [g + λH − k ∗ (λ)]∗ ⊂ epi (g + k ◦ H )∗ λ∈dom k ∗ Applying (5) with a = f and b = g + k ◦ H we get B ⊂ epi ( f + g + k ◦ H )∗ For the last assertion, let us first observe that A does not change when replacing f, g, k by f ∗∗ , g ∗∗ , and k ∗∗ Hence, we have A ⊂ epi f ∗∗ + g ∗∗ + k ∗∗ ◦ H ∗ ⊂ epi ( f + g + k ◦ H )∗ = A Consequently, the function ( f ∗∗ + g ∗∗ + k ∗∗ ◦ H )∗ and ( f + g + k ◦ H )∗ have the same epigraph and are equal The next proposition gives two sufficient conditions for having (I) They hold without any convexity, nor lower semi-continuity assumption on f, g, λH , and k ◦ H Proposition Given h ∈ Γ (X ), let us consider the statements (I I ) (I I I ) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃v1 ∈ dom g ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + g ∗ (v1 ) + (λH )∗ (w − v − v1 ) + k ∗ (λ) ≤ h ∗ (w) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + (g + λH )∗ (w − v) + k ∗ (λ) ≤ h ∗ (w) Then, we have (I I ) ⇒ (I I I ) ⇒ (I ) Proof • (I I ) ⇒ (I I I ) Just observe that (g + λH )∗ (w − v) ≤ g ∗ ✷(λH )∗ (w − v) ≤ g ∗ (v1 ) + (λH )∗ (w − v − v1 ) • (I I I ) ⇒ (I ) Let x ∈ A = dom f ∩ dom g ∩ H −1 (dom k) For any w ∈ dom h ∗ , by (III), one has x, w − h ∗ (w) ≤ x, w − f ∗ (v) − (g + λH )∗ (w − v) − k ∗ (λ) ≤ x, w − x, v + f (x) − x, w − v +g(x) + (λH )(x) − λ, H (x) + k(H (x)) = f (x) + g(x) + (k ◦ H )(x) 123 J Glob Optim Taking the supremum over w ∈ dom h ∗ , we get h(x) = h ∗∗ (x) ≤ f (x) + g(x) + (k ◦ H )(x) for any x ∈ A, and finally, f + g + k ◦ H ≥ h on X We now provide necessary and sufficient conditions ensuring respectively that the equivalence holds between (I ) and (I I ) and between (I ) and (III) In the sequel we just assume that f + g + k ◦ H is proper We begin with a lemma Lemma For any nonempty subset V of X ∗ , the following statements are equivalent: (i) (V × R) ∩ epi ( f + g + k ◦ H )∗ = (V × R) ∩ A, (ii) for each w ∈ V , ( f +g+k ◦ H )∗ (w) = λ∈domk ∗ v∈dom f ∗ ,v1 ∈dom g ∗ f ∗ (v) + g ∗ (v1 ) + (λH )∗ (w − v − v1 ) + k ∗ (λ) Proof (ii) means exactly that (w, r ) ∈ (V × R) ∩ epi ( f + g + k ◦ H )∗ if and only if w ∈ V and there exist (λ, v, v1 ) ∈ dom k ∗ × dom f ∗ × dom g ∗ such that f ∗ (v) + g ∗ (v1 ) + λH − k ∗ (λ) ∗ (w − v − v1 ) ≤ r, or, equivalently, if and only if (λ, v, v1 ) ∈ dom k ∗ × dom f ∗ × dom g ∗ , (s, t) ∈ R × R such that f ∗ (v) ≤ s, g ∗ (v1 ) ≤ t, (λH − k ∗ (λ))∗ (w − v − v1 ) ≤ r − s − t Thus, (V × R) ∩ epi ( f +g+k ◦ H )∗ = (V × R) ∩ epi f ∗ + epi g ∗ + epi (H −k ∗ (λ))∗ λ∈dom k ∗ = (V × R) ∩ A Analogously, we can state Lemma For any nonempty subset V of X ∗ , the following statements are equivalent: (i) (V × R) ∩ epi ( f + g + k ◦ H )∗ = (V × R) ∩ B, (ii) for each w ∈ V , ( f + g + k ◦ H )∗ (w) = λ∈domk ∗ v∈dom f ∗ f ∗ (v) + (g + λH )∗ (w − v) + k ∗ (λ) We are now in a position to prove the main results of this section More concretely, it is shown in the next theorems that relations like (2) or (3) are necessary and sufficient conditions for dual transcriptions of (I ) These results are considered as characterizations of Farkas-type results and also their stability under linear perturbations They serve as the main tools for establishing alternative type theorems, set containments (Sect 5), and for strong duality or stable strong duality for optimization problems (see Sect for concrete examples) Theorem The following assertions are equivalent (i) epi ( f + g + k ◦ H )∗ = A, 123 J Glob Optim (ii) For any h ∈ Γ (X ), (I ) ⇐⇒ (I I ), (iii) For any x ∗ ∈ X ∗ , any γ ∈ R, f + g + k ◦ H − x ∗ ≥ γ on X ∃v ∈ dom f ∗ , v1 ∈ dom g ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + g ∗ (v1 ) + (λH )∗ (x ∗ − v − v1 ) + k ∗ (λ) ≤ −γ Proof (i) ⇒ (ii) By Proposition 1, it is sufficient to check that (I ) ⇒ (I I ) So, let w ∈ dom h ∗ Observe firstly that if (i) holds then (i) in Lemma holds with V := dom h ∗ Secondly, if (I ) holds then one has ( f + g + k ◦ H )∗ (w) ≤ h ∗ (w) and hence, (w, h ∗ (w)) ∈ (V × R) ∩ epi ( f + g + k ◦ H )∗ = (V × R) ∩ A By (ii) of Lemma there exist v ∈ dom f ∗ , v1 ∈ dom g ∗ , λ ∈ dom k ∗ such that f ∗ (v) + g ∗ (v1 ) + (λH )∗ (w − v − v1 ) + k ∗ (λ) ≤ h ∗ (w), and (I I ) holds (ii) ⇒ (iii) For any x ∗ ∈ X ∗ and any γ ∈ R, take h(x) = x ∗ , x + γ for all x ∈ X in (ii) and then (iii) follows (iii) ⇒ (i) By Lemma we have just to check that the inclusion “⊂” holds in (i) Assume that ( f + g + k ◦ H )∗ (x ∗ ) ≤ r We thus have f + g + k ◦ H ≥ x ∗ − r on X Since (iii) holds, then there exist v ∈ dom f ∗ , v1 ∈ dom g ∗ , and λ ∈ dom k ∗ such that f ∗ (v) + g ∗ (v1 ) + (λH )∗ (x ∗ − v − v1 ) + k ∗ (λ) ≤ r, or equivalently, f ∗ (v) + g ∗ (v1 ) + (λH − k ∗ (λ))∗ (x ∗ − v − v1 ) ≤ r, which yields x ∗ − v − v1 ∈ dom (λH − k ∗ (λ))∗ Letting α := r − f ∗ (v) − g ∗ (v1 ) − (λH − k ∗ (λ))∗ (x ∗ − v − v1 ) ≥ 0, we get (x ∗ , r ) = (v, f ∗ (v)) + (v1 , g ∗ (v1 )) + x ∗ − v − v1 , (λH − k ∗ (λ))∗ (x ∗ − v − v1 ) + (0, α) ∈ epi f ∗ + epi g ∗ + epi (λH − k ∗ (λ))∗ + (0, α) ⊂ A, which is desired By the same way, using Lemma instead of Lemma we can state Theorem The following assertions are equivalent (i) epi ( f + g + k ◦ H )∗ = B, (ii) For any h ∈ Γ (X ), (I ) ⇐⇒ (I I I ), (iii) For any x ∗ ∈ X ∗ , any γ ∈ R, f +g+k ◦ H −x ∗ ≥ γ on X ⇐⇒ ∃v ∈ dom f ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + (g + λH )∗ (x ∗ − v) + k ∗ (λ) ≤ −γ When h is a constant function, we get the following corollaries from Theorems 2–3 and Lemmas 2–3 whose proofs are similar and only one of them is given Corollary The following statements are equivalent: (i) {0 X ∗ } × R 123 epi ( f + g + k ◦ H )∗ = {0 X ∗ } × R A, J Glob Optim (ii) For any γ ∈ R, f + g + k ◦ H ≥ γ on X ⇐⇒ ∃v ∈ dom f ∗ , ∃w ∈ dom g ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + g ∗ (w) + (λH )∗ (−v − w) + k ∗ (λ) ≤ −γ Corollary The following statements are equivalent: epi ( f + g + k ◦ H )∗ = {0 X ∗ } × R (i) {0 X ∗ } × R (ii) For any γ ∈ R, f + g + k ◦ H ≥ γ on X ⇐⇒ B, ∃v ∈ dom f ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + (g + λH )∗ (−v) + k ∗ (λ) ≤ −γ Proof (Proof of Corollary 2) (i) ⇒ (ii) We firstly observe that f + g + k ◦ H ≥ γ on X is equivalent to − γ ≥ ( f + g + k ◦ H )∗ (0 X ∗ ) (6) Since (i) holds, applying Lemma with V = {0 X ∗ }, one gets −γ ≥ λ∈domk ∗ v∈dom f ∗ f ∗ (v) + (g + λH )∗ (−v) + k ∗ (λ) , and (ii) follows (ii) ⇒ (i) Similar to the proof of (iii) ⇒ (i) in Theorem with h(x) = γ for all x ∈ X On the relations epi ( f + g + k ◦ H)∗ = A or B : the use of convexity and lower semicontinuity In this section, we will establish necessary and sufficient conditions for the relation epi ( f + g + k ◦ H )∗ = A or B and some new transcriptions of the functional inequality (I ) in the presence of convexity (resp convexity and lower semicontinuity) As an application of these transcriptions, we consider some concrete situation where these conditions characterize Farkas-type results and/or stable form of these results, stable Lagrange duality for convex optimization problems with convex conic constraints (Corollaries 4, 5) Through out this section, we assume that f, g, k are proper convex functions 4.1 A first approach Assume that f, g ∈ Γ (X ), k ∈ Γ (Z ), and λH ∈ Γ (X ), ∀λ ∈ dom k ∗ (7) Observe that (7) is in particular satisfied if k is S-increasing, H is S-convex and star-S-lower semicontinuity (i.e., λH is lsc for all λ ∈ S ∗ ) Let us now introduce the extended real-valued functions ϕ and ψ defined on X ∗ by ϕ= ψ = inf f ∗ ✷g ∗ ✷[λH − k ∗ (λ)]∗ , inf f ∗ ✷[g + λH − k ∗ (λ)]∗ λ∈dom k ∗ λ∈dom k ∗ 123 J Glob Optim By using classical arguments, one easily sees that ϕ and ψ are convex and ϕ∗ = f + g + k ◦ H = ψ ∗ Assuming moreover f + g + k ◦ H is proper (i.e., dom f ∩ dom g ∩ H −1 (dom k) = ∅), we have w ∗ − cl epi ϕ = epi ( f + g + k ◦ H )∗ = w ∗ − cl epi ψ Denoting by epi s ϕ the strict epigraph of ϕ, one also has epi s ϕ = epi s f ∗ + epi s g ∗ + epi s [λH − k ∗ (λ)]∗ , λ∈dom k ∗ epi s ψ = epi s f ∗ + epi s [g + λH − k ∗ (λ)]∗ λ∈dom k ∗ Taking the w ∗ -closure in the above relations we get easily w ∗ − cl A = w ∗ − cl epi ϕ = w ∗ − cl epi ψ = w ∗ − cl B, and we can state that: Theorem Assume that f + g + k ◦ H is proper Then epi ( f + g + k ◦ H )∗ = A (resp B) if and only if A (resp B) is w ∗ -closed 4.2 Using Moreau-Rockafellar theorem In this subsection we assume that k is S-increasing on H (dom H ) + S and H is S-convex Let us slightly modify the set A and B by setting A : = epi f ∗ + epi g ∗ + epi λH − k ∗ (λ) ∗ , λ∈dom k ∗ ∩S ∗ B : = epi f ∗ + epi g + λH − k ∗ (λ) ∗ , λ∈dom k ∗ ∩S ∗ and, for h ∈ Γ (X ), the corresponding statements (I I ) (I I I ) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃v1 ∈ dom g ∗ , ∃λ ∈ dom k ∗ ∩ S ∗ , f ∗ (v) + g ∗ (v1 ) + (λH )∗ (w − v − v1 ) + k ∗ (λ) ≤ h ∗ (w) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃λ ∈ dom k ∗ ∩ S ∗ , f ∗ (v) + (g + λH )∗ (w − v) + k ∗ (λ) ≤ h ∗ (w) It is worth noticing that if k is S-increasing on the whole Z , then dom k ∗ ⊂ S ∗ , A = A , B = B , and (I I ) ≡ (I I ) , (I I I ) ≡ (I I I ) As repetitions of Theorems 2, we can state: Theorem The following assertions hold: epi ( f + g + k ◦ H )∗ = A ⇐⇒ (I ) ⇔ (I I ) , ∀h ∈ Γ (X ) , epi ( f + g + k ◦ H )∗ = B ⇐⇒ (I ) ⇔ (I I I ) , ∀h ∈ Γ (X ) Remark It should be emphasized that, by a reasoning similar to that of Theorem 3, Theorem still holds when replacing “∀h ∈ Γ (X )” by “∀x ∗ ∈ X ∗ , ∀γ ∈ R” In the same way, when h are constant functions, the following corollary is at hand, whose proof is similar to those of Corollaries 1–2 and will be omitted 123 J Glob Optim Applying [2, Theorem 9.2] with n = 3, = Fi , we obtain that (ii) is equivalent to (for all w ∈ Δ) (F1 + F2 + F3 )∗ (w, Z ∗ ) = ∗ F1∗ (v, λ) + F2∗ (v1 , μ) + F3∗ (w − v − v1 , −λ − μ) v,v1 ∈X λ,μ∈Z ∗ Now, the right hand side of the previous equality is equal to v∈dom f ∗ v1 ∈dom g ∗ λ∈dom k ∗ ∩S ∗ f ∗ (v) + g ∗ (v1 ) + (λH )∗ (w − v − v1 ) + k ∗ (λ) = v∈dom f ∗ v1 ∈dom g ∗ λ∈S ∗ f ∗ (v) + g ∗ (v1 ) + (λH )∗ (w − v − v1 ) + k ∗ (λ) and since k is S-increasing on H (dom H ) + S, one has (F1 + F2 + F3 )∗ (w, Z ∗ ) = ( f + g + k ◦ H )∗ (w) It follows that (w, r ) belongs to Δ × R ∩ epi ( f + g + k ◦ H )∗ if and only if there exist v ∈ dom f ∗ , v1 ∈ dom g ∗ , λ ∈ dom k ∗ ∩ S ∗ , s, s1 , t ∈ R such that f ∗ (v) ≤ s, g ∗ (v1 ) ≤ s1 , (λH )∗ (w − v − v1 ) ≤ t, s + s1 + t = r , or, equivalently, (v, s) ∈ epi f ∗ , (v1 , s1 ) ∈ epi g ∗ , (w − v − v1 , t) ∈ epi (λH − k ∗ (λ)) , s + s1 + t = r Hence, (ii) amounts to Δ × R ∩ epi ( f + g + k ◦ H )∗ = epi f ∗ + epi g ∗ + epi (λH − k ∗ (λ))∗ λ∈dom k ∗ ∩S ∗ =A Theorems 8–9, combining with Theorem and Corollary 3, may lead to complete characterizations of Farkas-type results for convex/DC systems, Lagrange duality for convex problems under convex conic constraints and their stable form by closedness conditions which improve the known ones in the literature These are given in the next corollaries However, we first introduce some other closedness conditions Let us take a closer look to the general convex conic constraint problem: (PC) inf x∈C,H (x)∈−S f (x) and its perturbed version (for any x ∗ ∈ X ∗ ) (PC x ∗ ) inf x∈C,H (x)∈−S f (x) − x ∗ , x , where H is S-convex and S-epi-closed, f ∈ Γ (X ), and C is a closed convex subset of X Assume that dom f ∩ C ∩ H −1 (−S) = ∅ Let us introduce the mapping F(x, z) := f (x) + i C (x) + i epi H (x, z) One has F ∈ Γ (X × Z ) and F ∗ (w, −λ) = ( f + i C + λH )∗ (w) +∞ if λ ∈ S ∗ , otherwise 123 J Glob Optim For w ∈ X ∗ , defining p(w) := inf λ∈S ∗ F ∗ (w, −λ), it holds p ∗ (x) = F ∗∗ (x, Z ) = F(x, Z ) In other words, p ∗ (x) = f (x) +∞ if x ∈ C and H (x) ∈ −S, otherwise Therefore, inf(PC) = inf p ∗ = − p ∗∗ (0 X ∗ ) and inf(PC x ∗ ) = − p ∗∗ (x ∗ ) Now, since p is X convex and dom p ∗ = ∅, p ∗∗ = w ∗ − cl p, and hence, for any x ∗ ∈ X ∗ and any γ ∈ R we have inf(PC) ≥ γ ⇐⇒ (w ∗ − cl p)(0 X ∗ ) ≤ −γ ⇐⇒ (0 X ∗ , −γ ) ∈ w ∗ − cl(epi p), ∗ and ∗ inf(PC ) ≥ γ ⇐⇒ (x , −γ ) ∈ w − cl(epi p) x∗ On the other hand, w ∗ − cl(epi p) = w ∗ − cl(epi s p) = w ∗ − cl epi s ( f + i C + λH )∗ λ∈S ∗ = w ∗ − cl epi ( f + i C + λH )∗ = w ∗ − clC , λ∈S ∗ where C = λ∈S ∗ epi ( f + i C + λH )∗ Consequently, for any x ∗ ∈ X ∗ and any γ ∈ R, inf(PC) ≥ γ ⇐⇒ (0 X ∗ , −γ ) ∈ w ∗ − cl C , inf(PC x ∗ ) ≥ γ ⇐⇒ (x ∗ , −γ ) ∈ w ∗ − cl C , (8) (9) The last equivalences (i.e., (8) and (9)) enable us to introduce new closedness conditions as shown in the next corollaries Corollary Let H be S-convex and S-epi-closed, f ∈ Γ (X ), and C be a closed convex subset of X Assume that dom f ∩ C ∩ H −1 (−S) = ∅ The following statements are equivalent: (i) {0 X ∗ } × S ∗ × R+ + λ∈S ∗ (w, −λ, r ) | (w, r ) ∈ epi ( f + i C + λH )∗ is w ∗ -closed regarding to the set X ∗ × {0 Z ∗ } × R, ∗ ∗ ∗ (ii) λ∈S ∗ epi ( f + i C + λH ) is w -closed regarding to the set X × R, (iii) (Farkas lemma with convex and DC functions) For any h ∈ Γ (X ), x ∈ C, H (x) ∈ −S ⇒ f (x)−h(x) ≥ ⇐⇒ ∀w ∈ dom h ∗ , ∃λ ∈ S ∗ , ( f 0∗ + i C + λH )∗ (w) ≤ h ∗ (w) , (iv) (Stable Farkas lemma for convex systems) For any x ∗ ∈ X ∗ and any γ ∈ R, x ∈ C, H (x) ∈ −S ⇒ f (x)− x ∗ , x ≥ γ ⇐⇒ ∃λ ∈ S ∗ , f + λH −x ∗ ≥ γ on C , (v) (Stable Lagrange duality) For any x ∗ ∈ X ∗ , inf x∈C,H (x)∈−S 123 f (x) − x ∗ , x = max∗ inf λ∈S x∈C f (x) + λH (x) − x ∗ , x J Glob Optim Proof Firstly, let us apply Theorem with f ≡ 0, g = f + i C , k = i {−S} , and Δ = X ∗ Then k is S-increasing on X, k ∗ = i S ∗ , and one has epi ( f ∗ ⊕ i S ∗ ) = {0 X ∗ } × S ∗ × R+ , epi ( f + i C + λH )∗ B = {0 X ∗ } × R+ + λ∈S ∗ Thus, Theorem ensures that (i) is equivalent to epi ( f + i C + k◦H )∗ = B The equivalence between (i) and (iii) now follows from Theorem It also follows from Theorem that (i) is equivalent to: ∀x ∗ ∈ X ∗ , ∀γ ∈ R, x ∈ C, H (x) ∈ −S ⇒ f (x) + i C (x) − x ∗ , x ≥ γ ∃λ ∈ S ∗ , ( f + i C + λH )∗ (x ∗ ) ≤ −γ which is nothing but (iv) Finally, as dom ( f − x ∗ ) ∩ C ∩ H −1 (−S) = ∅, inf x∈C,H (x)∈−S f (x) − x ∗ , x < +∞ and hence, the equivalence between (iv) and (v) is straightforward Secondly, we show that (ii) is equivalent to (iv) This follows from (9) Indeed, if (ii) holds then the left hand side of (iv) means that inf(PC x ∗ ) ≥ γ By (ii) and (9), this is equivalent to (x ∗ , −γ ) ∈ C which means that there is λ ∈ S ∗ such that the right hand side of (iv) holds Conversely, if (iv) holds and (x ∗ , −γ ) ∈ w ∗ − cl C for some x ∗ ∈ X ∗ Since w ∗ − cl C = epi p ∗∗ , one has inf(PC x ∗ ) = − p ∗∗ (x ∗ ) ≥ −γ By (iv) this is equivalent to the fact that there exists λ ∈ S ∗ such that f + λH − x ∗ ≥ γ on C which means that (x ∗ , −γ ) ∈ C (ii) has been proved and we are done Remark The equivalences [(i) ⇐⇒ (iii)] and [(ii) ⇐⇒ (iii)] in Corollary characterize Farkas lemma for systems involving DC functions which strengthen the results in [8, Theorem 4.6], where only a sufficient condition for (iii) is proposed The other equivalences between (i) or (ii) and (iv) give complete characterizations of stable Farkas lemma which extend the one in [15, Theorem 3.1] where C = X and the mapping H was assumed to be continuous while [(i) ⇐⇒ (v)] and [(ii) ⇐⇒ (v)] are the characterizations of stable Lagrange duality which extends [24, Corollary 2.7.3], and may be derived from [2, Theorem 9.1] with some suitable choices of the perturbation function Φ and the subspace V of X ∗ Corollary Let H, f , and C be the same as in Corollary Assume that dom f ∩ C ∩ H −1 (−S) = ∅ The following statements are equivalent: (i) {0 X ∗ } × S ∗ × R+ + λ∈S ∗ (w, −λ, r ) | (w, r ) ∈ epi ( f + i C + λH )∗ is w ∗ -closed regarding to the set {0 X ∗ } × {0 Z ∗ } × R, ∗ ∗ (ii) λ∈S ∗ epi ( f + i C + λH ) is w -closed regarding to the set {0 X ∗ } × R, (iii) (Farkas lemma for convex systems) For any γ ∈ R, x ∈ C, H (x) ∈ −S ⇒ f (x) ≥ γ ⇐⇒ ∃λ ∈ S ∗ , ∀x ∈ C, f (x) + (λH )(x) ≥ γ , (iv) (Lagrange duality) inf x∈C,H (x)∈−S f (x) = max∗ inf λ∈S x∈C f (x) + λH (x) 123 J Glob Optim Proof It follows from Theorem (with Δ = {0 X ∗ }, f ≡ 0, g = f + i C , k = i {−S} ) and Corollary that (i) is equivalent to x ∈ C, H (x) ∈ −S ⇒ f (x) ≥ γ ∃λ ∈ S ∗ , ( f + i C + λH )∗ (0 X ∗ ) ≤ −γ , which, in turn, is equivalent to (iii) The equivalence of (iii) and (iv) is obvious The proof of the equivalence between (ii) and (iii) is similar to that of [(ii) ⇔ (iv)] of Corollary 4, using (8) instead of (9) The proof is complete Remark Similar to Corollary 4, Corollary gives a characterization of Farkas lemma for cone convex systems It extends the one in [5, Theorem 2.2] where only a sufficient condition was proposed and for the special case C = X It is also worth mentioning that this kind of results (characterization for Farkas-type results), due to the best knowledge of the authors, appeared in the literature for the first time in [14, Theorem 3.1] and the result established for a special case where C = X, H is continuous, S-convex, and f is convex, continuous On the other hand, the equivalence between (i) and (iv) (or (ii) and (iv)) gives a characterization for Lagrange duality which extends the corresponding result in [13, Theorem 3.1] where H is S-convex and continuous, and f is continuous convex function This extends [24, Theorem 2.7.1] and may be derived from [2, Theorem 9.1] with suitable choices of the perturbation function Φ and the subspace V of X ∗ Alternative type theorems In this section we exploit Theorems 2, (and other results in Sect 4) in order to obtain alternative type theorems, including the ones involving DC vector-valued mappings These results are new in two features: firstly, they are rather general in the sense that they involve composite functions, there are no assumptions on continuity, nor on convexity imposed on the data, and covers many other ones established recently in the literatures for convex/DC systems; secondly, they supply necessary and sufficient conditions for alternative type theorems while most of similar results appeared in the literature concerned only the sufficient conditions (see [12,17] and references therein) 5.1 Alternative theorems involving composite functions in general setting Let A := dom f ∩ dom g ∩ H −1 (dom k) = dom ( f + g + H ◦ k) Theorem 10 Let f, g : X → R ∪ {+∞}, k : Z → R ∪ {+∞} be proper and H : dom H ⊂ X → Z with dom H = ∅ Then, epi ( f + g + k ◦ H )∗ = A if and only if for any h ∈ Γ (X ), precisely one of the following statements is true (i) ∃x ∈ A, f (x) + g(x) + k[H (x)] < h(x), (ii) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃v1 ∈ dom g ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + g ∗ (v1 ) + (λH )∗ (w − v − v1 ) + k ∗ (λ) ≤ h ∗ (w) Proof Observe that non-(i) is equivalent to (I ) and (ii) is nothing but (I I ) Then, Theorem yields the result 123 J Glob Optim Theorem 11 With the hypothesis of Theorem 10, epi ( f + g + k ◦ H )∗ = B if and only if for any h ∈ Γ (X ), precisely one of the following statements is true (i) ∃x ∈ A, f (x) + g(x) + k[H (x)] < h(x), (ii) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃λ ∈ dom k ∗ , f ∗ (v) + (g + λH )∗ (w − v) + k ∗ (λ) ≤ h ∗ (w) Proof Observe that non-(i) is equivalent to (I ) and (ii) is nothing but (I I I ) Then, Theorem yields the result 5.2 Alternative theorems involving convex, DC functions, and conic constraints In this subsection, we will show that the alternative theorems just established (Theorems 10–11) extend the Farkas-type results involving cone-convex and DC functions established recently in [8,9] Let f ∈ Γ (X ), C be a closed convex subset of X , and Z be preordered by a non-negative, closed convex cone ∅ = S ⊂ Z Let further k := i {−S} and g := i C Then g ∈ Γ (X ) and k ∈ Γ (Z ) and k is increasing with respect to S Moreover, assume that H is S-convex and A = dom f ∩ C ∩ H −1 (−S) = ∅ 5.2.1 Star-S-lower semi-continuous conic constraints We now assume further that the mapping H is star-S-lower semi-continuous, meaning that λ ∈ S∗ ⇒ λH is lower semi-continuous In this setting, as it is shown in Sect (Theorem 4) that A = epi f ∗ + λ∈S ∗ epi (λH )∗ + epi i C∗ and the condition epi ( f + i C + i {−S} ◦ H )∗ = A holds if and only if epi f ∗ + ∗ ∗ ∗ λ∈S ∗ epi (λH ) + epi i C is weak -closed, which is the condition (CC) in [8,9] Theorem 12 Assume that f, C, H, S are as above with A = ∅ Then the following statement are equivalent: (a) The set epi f ∗ + λ∈S ∗ epi (λH )∗ + epi i C∗ is weak ∗ -closed, (b) For any h ∈ Γ (X ), precisely one of the following statements is true: (i) ∃x ∈ C : H (x) ∈ −S and f (x) < h(x), (ii) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃v1 ∈ dom i C∗ , ∃λ ∈ S ∗ , f ∗ (v) + i C∗ (v1 ) + (λH )∗ (w − v − v1 ) ≤ h ∗ (w) ∗ = Proof This is a direct consequence of Theorem 10, just observe that dom k ∗ = dom i {−S} S∗ It worth observing that a similar result to Theorem 12 can be obtained if the condition epi ( f + g + k ◦ H )∗ = B is used instead of epi ( f + g + k ◦ H )∗ = A The assumption that H is star-S- semi-continuous will be relaxed in the next subsection 5.2.2 S-epi-closed conic constraints We intend to apply Theorem to the case when k = i {−S} and g = i C We will assume that H : dom H −→ Z is S-epi-closed, meaning that epi H := {(x, z) ∈ dom H × Z | z − H (x) ∈ S} 123 J Glob Optim is closed in X × Z We now can establish Theorem 13 Assume that f, C, H, S are as above with A = ∅ Then the following statements are equivalent: (a) The set λ∈S ∗ {(v, λ, r ) | f ∗ (v) ≤ r } + λ∈S ∗ {(w, −λ, r ) | (i C + λH )∗ (w) ≤ r } is weak∗ -closed regarding to the set X ∗ × {0 Z ∗ } × R, (b) For any h ∈ Γ (X ), precisely one of the following statements is true: (i) ∃x ∈ C : H (x) ∈ −S and f (x) < h(x), (ii) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃λ ∈ S ∗ : f ∗ (v) + (i C + λH )∗ (w − v) ≤ h ∗ (w) Proof Let us first observe that B = B Thus, Theorem (with Δ = X ∗ ) says that (a) is equivalent to B = epi ( f + g + k ◦ H )∗ We conclude the proof by applying Theorem 11 5.2.3 Using slater condition Theorem 14 Assume that f is convex, C is convex, H is S-convex and there exists x0 , x1 ∈ C ∩ dom f ∩ dom H such that H (x0 ) ∈ −intS and f is continuous at x1 Then the statement (b) in Theorem 13 holds Proof By Theorem (with g = i C and k = i {−S} ) one has epi ( f + i C + i {−S} ◦ H )∗ = B = B We can conclude the proof by using Theorem 11 As mentioned above, Theorems 10–11 have many applications Here we just focus our attention on the following extended versions of the Farkas Lemma Corollary Let T : X → Z be linear and continuous, C ⊂ X and D ⊂ Z be two convex sets such that T (C) ∩ int D = ∅ (10) Then, for any h ∈ Γ (X ), precisely one of the following statements is true: (i) ∃x ∈ C, T x ∈ D and h(x) > 0, ∗ (λ) ≤ h ∗ (w), where b(D) denotes the (ii) ∀w ∈ dom h ∗ , ∃λ ∈ b(D), i C∗ (w − T ∗ (λ)) + i D ∗ barrier cone of D and T denotes the adjoint of the linear operator T Proof We first apply Theorem 14 with f ≡ 0, g = i C , k = i D , H = T , and S = {0 Z } Thus, S ∗ = Z ∗ and B = B By (10), k is continuous at a point H (x0 ) with x0 ∈ dom f ∩ dom g ∩ H −1 (dom k), and f is of course continuous at x0 Thus, Theorem (with Δ = X ∗ ) yields B = B = epi (i C +i D ◦ T )∗ Now Theorem 11 says that precisely one of the following statements is true: (i) ∃x ∈ C, T x ∈ D and h(x) > 0, ∗ (λ) ≤ h ∗ (w) (ii) ∀w ∈ dom h ∗ , ∃λ ∈ b(D), (i C + λT )∗ (w) + i D Since (i C + λT )∗ (w) = i C∗ (w − T ∗ (λ)), we are done Corollary Let T : X → Z be linear and continuous, z ∈ Z , and P ⊂ X, Q ⊂ Z be two convex cones such that T (P) ∩ int(z − Q) = ∅ Then, for any h ∈ Γ (X ), precisely one of the following statements is true: 123 J Glob Optim (i) ∃x ∈ P, z − T (x) ∈ Q and h(x) > 0, (ii) ∀w ∈ dom h ∗ , ∃λ ∈ Q ∗ , T ∗ (λ) − w ∈ P ∗ and λ, z ≤ h ∗ (w) ∗ = ·, z + Proof Apply Corollary with C = P and D = z − Q Then i C∗ = i {−P ∗ } , i D i Q ∗ , b(D) = Q ∗ , and the result follows It is worth observing that the alternative theorems proposed above may give rise to set containment results (see, e.g., [12,17]) As an example, we just focus on the concrete system x ∈ C ∩ H −1 (S), f (x) < h(x) This system is inconsistent if and only if f + i C + i {−S} ◦ H ≥ h Let [H ≤ S Z ], [ f ≥ h] denote the sets {x ∈ X | H (x) ∈ −S} and {x ∈ X | f (x) ≥ h(x)}, respectively Then the last inequality can also be written as C ∩ [H ≤ S Z ] ⊂ [ f ≥ h], which is a containment of a convex set in a DC set [21] (the set defined by a difference of two convex functions, here we mean [ f − h ≥ 0]) In case h ≡ 0, the mentioned containment collapses to the one of a convex set in a reverse convex set (see [12]): C ∩ [H ≤ S Z ] ⊂ [ f ≥ 0] (11) As an example, we apply Theorem 11 to the present situation Proposition Assume that C ⊂ X is closed and convex, f ∈ Γ (X ), and H : dom H → Z is S-convex and continuous, where S ⊂ Z is a closed convex cone If moreover int(−S) ∩ H (C ∩ dom H ) = ∅, f is finite and continuous at a point of C ∩ [H ≤ Z ] then, for any h ∈ Γ (X ), the following statements are equivalent (i) C ∩ [H ≤ S Z ] ⊂ [ f ≥ h], (ii) ∀w ∈ dom h ∗ , ∃v ∈ dom f ∗ , ∃λ ∈ S ∗ , f ∗ (v) + (i C + λH )∗ (w − v) ≤ h ∗ (w) Proof Apply Theorem 11 with g = i C and k = i {−S} 5.3 Alternative type theorems involving DC vector-valued mappings Given C ⊂ X, f, h : X → R ∪ {+∞}, S ⊂ Z a convex cone such that −S ∩ S = {0 Z }, Hi : dom Hi ⊂ X → Z (i=1,2) and D := C ∩ dom H1 ∩ dom H2 Let us consider the problem (P) Find x ∈ D such that H1 (x) ≤ S H2 (x) and f (x) < h(x) In order to discuss an alternative statement for (P), let us introduce the S-subdifferential of the mapping H2 : dom H2 ⊂ X → Z at x ∈ dom H2 by setting (see [19,20,22,23] and the references therein) ∂ S H2 (x) := {L ∈ L(X, Z ) | H2 (u) − H2 (x) ≥ S L(u − x), ∀u ∈ dom H2 } One has, for any L ∈ ∂ S H2 (x), L(x) − H2 (x) = max [L(u) − H2 (u)] =: H2∗ (L), u∈dom H2 123 J Glob Optim where the maximum is taken with respected to the order relation “≤ S ” This defines H2∗ : dom H2∗ = u∈dom H2 ∂ S H2 (u) → Z as the exact conjugate of H2 (see [20]) Then Fenchel inequality L(x) − H2 (x) ≤ S H2∗ (L) (12) holds for any x ∈ dom H2 and any L ∈ dom H2∗ For any L ∈ dom H2∗ , let us introduce the sub-problem (PL ) Find x ∈ D, such that H1 (x) − L(x) + H2∗ (L) ≤ S Z and f (x) < h(x), and define HL (x) := H1 (x) − L(x) + H2∗ (L), ∀x ∈ dom H1 ∩ dom H2 (13) Lemma Assume that H2 is S-subdifferentiable on D The following assertions are equivalent: (i) (P) is unfeasible, (ii) (PL ) is unfeasible for any L ∈ dom H2∗ , (iii) ∀L ∈ Δ := ∂ S H2 (D), f + i C + i {−S} ◦ HL ≥ h on X Proof • (i) implies (ii) is due to (12) • (ii) implies (iii) comes by the definition (13) of HL (observe that dom HL = dom H1 ∩ dom H2 ) • (iii) implies (i) Let x ∈ D be such that H1 (x) ≤ S H2 (x) and let us prove that f (x) ≥ h(x) Pick L ∈ ∂s H2 (x) ⊂ Δ One has H1 (x) ≤ S H2 (x) = L(x) − H2∗ (L), i.e., HL (x) ≤ S Z and, thanks to (iii), f (x) ≥ h(x) The results of Sects and provide alternative formulations for each sub-problem (PL ) We choose here to apply Theorem 14 with H = HL , g = i C and k = i {−S} (and hence, k ∗ = i S ∗ ) Theorem 15 Let C ⊂ X, f : X → R ∪ {+∞} be convex, S ⊂ Z a convex cone such that −S ∩ S = {0 Z }, H1 : dom H1 ⊂ X → Z , H2 : dom H2 ⊂ X → Z Assume that H1 is S-convex, H2 is S-subdifferentiable on D = C ∩ dom H1 ∩ dom H2 Assume moreover that, for any L ∈ Δ = ∂s H2 (D), (i) f is finite and continuous at a point of {x ∈ D | H1 (x) − L(x) + H2∗ (L) ≤ S Z }, (ii) ∃x L ∈ D, H1 (x L ) − L(x L ) + H2∗ (L) ∈ int(−S) Then, for any h ∈ Γ (X ), precisely one of the following statements is true: (a) ∃x ∈ D, H1 (x) ≤ S H2 (x) and f (x) < h(x), (b) ∀L ∈ Δ, ∀w ∈ dom h ∗ , ∃λ ∈ S ∗ such that f ∗ (v) + (i C + λH1 )∗ (w − v + L ∗ (λ)) − λ, H2∗ (L) ≤ h ∗ (w), where L ∗ denotes the adjoint operator of L Proof This is a consequence of Lemma and Theorem 14 123 J Glob Optim Applications to optimization problems and Farkas-type results involving composite functions The use of “double composite model” as in Fig (Sect 1) enables us to use the results in the previous sections to get new Farkas-type results for variant systems of convex/nonconvex inequalities In turns, these new results give rise to duality results and optimality conditions for optimization problems involving composite functions As an illustration of the idea, in this section, we consider only two special circumstances with convex settings Let X, U, V be l.c.H.t.v.s., M : dom M ⊂ X → U, N : dom N ⊂ X → V be mappings and p : U → R ∪ {+∞}, q : V → R ∪ {+∞} be proper functions Let further Z := U × V and H : X → Z • , k : Z → R ∪ {+∞} be the mappings defined as H (x) := (M(x), N (x)) for all x ∈ dom H = dom M ∩ dom N and H (x) = ∞ Z if x ∈ dom H , and k(z) = p(u) + q(v) for all z = (u, v) ∈ Z , respectively We set, by convention, k(∞ Z ) = +∞ Therefore, (k ◦ H )(x) = p ◦ M(x) + q ◦ N (x) if x ∈ dom H and (k ◦ H )(x) = +∞ if x ∈ dom H Assume that U+ and V+ are closed convex cones in U and V , respectively Set S = U+ × V+ Then it is clear that S ∗ = U+∗ × V+∗ By this way, the Fig in Sect becomes the “normal” composite model shown in Fig and then the results obtained in the previous sections can be applied Here, two special cases are considered and several topological closedness conditions (qualification conditions) are introduced to characterize strong duality (of optimization problems) and Farkas-type results for systems in consideration 6.1 Strong duality and stable Farkas lemma involving convex composite functions In this subsection, we introduce some closedness conditions which characterize Farkas lemma (stable Farkas lemma) for systems involving composite convex functions and stable duality for composite convex problems We retain the following four assumptions: (α1 ) M is U+ -convex and U+ -epi-closed; N is V+ -convex and V+ -epi-closed, (α2 ) p ∈ Γ (U ) and q ∈ Γ (V ), (α3 ) ∃x ∈ dom M ∩ dom N , p(M(x)) < +∞ and q(N (x)) < +∞, p is U+ − increasing on M(dom M) + U+ (α4 ) q is V+ − increasing on N (dom N ) + V+ , Then it follows from (α1 ) − (α4 ) that k ∈ Γ (Z ), H is S-convex and S-epi-closed, k ◦ H is proper, and k is S-increasing on H (dom H ) + S, which means that all the assumptions in Sect 4.3 are satisfied (with f ≡ and g ≡ 0) We get from Theorem and Theorem the following results of Farkas-type (and the stable ones) involving composite convex and DC functions Theorem 16 The following statements are equivalent: Fig Composite model 123 J Glob Optim (i) The following set is w ∗ -closed regarding the set X ∗ × {0U ∗ } × {0V ∗ } × R: {0 X ∗ } × epi ( p ∗ ⊕ q ∗ ) (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) ∀h ∈ Γ (X ), ( p ◦ M)(x) + (q ◦ N )(x) ≥ h(x), ∀x ∈ X ∀x ∗ ∈ dom h ∗ , ∃ (u ∗ , v ∗ ) ∈ (U+∗ × V+∗ ) ∩ (dom p ∗ × dom q ∗ ), (u ∗ ◦ M + v ∗ ◦ N )∗ (x ∗ ) + p ∗ (u ∗ ) + q ∗ (v ∗ ) ≤ h ∗ (x ∗ ) (iii) ∀x ∗ ∈ X ∗ , ∀γ ∈ R, ( p ◦ M)(x) + (q ◦ N )(x) − x ∗ (x) ≥ γ , ∀x ∈ X ∃ (u ∗ , v ∗ ) ∈ (U+∗ × V+∗ ) ∩ (dom p ∗ × dom q ∗ ), (u ∗ ◦ M + v ∗ ◦ N )∗ (x ∗ ) + p ∗ (u ∗ ) + q ∗ (v ∗ ) ≤ −γ Proof With our notations and f ≡ 0, g ≡ 0, (i) is nothing but Theorem 8(ii) and hence, by Theorem 8, this condition is equivalent to epi (k ◦ H )∗ = epi ( p ◦ M + q ◦ N )∗ = B The equivalence [(i) ⇔ (ii)] follows from Theorem while [(i) ⇔ (iii)] is a consequence of the mentioned theorem and Remark The same argument, using Theorem and Corollary we get the following version of Farkas lemma involving composite convex functions Corollary The following statements are equivalent: (i) The following set is w ∗ -closed regarding the set {0 X ∗ } × {0U ∗ } × {0V ∗ } × R: {0 X ∗ } × epi ( p ∗ ⊕ q ∗ ) (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) ∀γ ∈ R, ( p ◦ M)(x) + (q ◦ N )(x) ≥ γ , ∀x ∈ X ∃ (u ∗ , v ∗ ) ∈ (U+∗ × V+∗ ) ∩ (dom p ∗ × dom q ∗ ) : u ∗ ◦ M(x) + v ∗ ◦ N (x) − p ∗ (u ∗ ) − q ∗ (v ∗ ) ≥ γ , ∀x ∈ X 123 J Glob Optim We now consider a general optimization problem involving convex composite functions: (P1) inf p ◦ M(x) + q ◦ N (x) x∈X The dual problem (D1) of (P1) is defined as sup inf u ∗ ◦ M + v ∗ ◦ N − p ∗ (u ∗ ) − q ∗ (v ∗ ) ∗ X u ∗ ∈dom p ∗ ∩U+ v ∗ ∈dom q ∗ ∩V+∗ It is easy to check that the weak duality holds, i.e., inf(P1) ≥ sup(D1) Indeed, for all x ∈ X and any u ∗ ∈ U+∗ ∩ dom p ∗ , v ∗ ∈ V+∗ ∩ dom q ∗ , one has u ∗ ◦ M(x) + v ∗ ◦ N (x) − p ∗ (u ∗ ) − q ∗ (v ∗ ) ≤ u ∗ ◦ M(x) + v ∗ ◦ N (x) + p(M(x)) − u ∗ , M(x) + q(N (x)) − v ∗ , N (x) = p ◦ M(x) + q ◦ N (x), (14) and the weak duality follows We will see now that the closedness condition (i) in Theorem 16 also characterizes stable strong duality for (P1) as well Theorem 17 The following statements are equivalent: (i) The following set is w ∗ -closed regarding the set X ∗ × {0U ∗ } × {0V ∗ } × R: {0 X ∗ } × epi ( p ∗ ⊕ q ∗ ) (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) Stable strong duality holds between (P1) and (D1), i.e., for all x ∗ ∈ X ∗ , inf p ◦ M +q ◦ N −x ∗ = x∈X max inf u ∗ ◦ M +v ∗ ◦ N − p ∗ (u ∗ )−q ∗ (v ∗ )−x ∗ ∗ x∈X u ∗ ∈dom p ∗ ∩U+ v ∗ ∈dom q ∗ ∩V+∗ Proof [(i) ⇒ (ii)] By (α3 ), dom ( p ◦ M + q ◦ N − x ∗ ) = ∅ and hence, γ := inf x∈X p ◦ M + q ◦ N − x ∗ < +∞ The assertion (ii) now follows from Theorem 16(iii) and the weak duality (see (14)) [(ii) ⇒ (i)] If (ii) holds then Theorem 16(iii) holds This ensures that (i) of Theorem 16 holds which is desired In the same way, the closedness condition Corollary 8(i) characterizes strong duality for (P1) as given in the next corollary whose proof is easy and will be omitted Corollary The following statements are equivalent: (i) The following set is w ∗ -closed regarding the set {0 X ∗ } × {0U ∗ } × {0V ∗ } × R: {0 X ∗ } × epi ( p ∗ ⊕ q ∗ ) (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) Strong duality holds between (P1) and (D1), i.e., inf p ◦ M + q ◦ N = X max inf u ∗ ◦ M + v ∗ ◦ N − p ∗ (u ∗ ) − q ∗ (v ∗ ) ∗ X u ∗ ∈dom p ∗ ∩U+ v ∗ ∈dom q ∗ ∩V+∗ 123 J Glob Optim 6.2 Convex composite problem with convex conic constraint We maintain the spaces X, U, V , the closed convex cones U+ , V+ , the mappings M, N , and the function q as in Sect 6.1 and let p := i {−U+ } We maintain also the assumptions (α1 ) − (α4 ) It is worth mentioning that p = i {−U+ } satisfies (α2 ), (α4 ), and the condition (α3 ) means that there exists x ∈ U ∩ dom N such that M(x) ∈ −U+ and q(N (x)) < +∞ Moreover, in this case, dom p ∗ = U+∗ and p is U+ -increasing on the whole space X In this concrete setting, the Problem (P1) collapses to the convex composite problem under a conic constraint (P2) below (P2) inf M(x)∈−U+ q ◦ N (x) We now consider a more general problem: the DC problem involving composite functions (P3) below Let h be some function belonging to Γ (X ) (P3) inf M(x)∈−U+ q ◦ N (x) − h(x) The “composite-DC” problem of the model (P3) appears in many real world nonconvex optimization problems (see, e.g., [1]) In the case where V = R and q = IdR , (P3) collapses to the usual DC problem which is one of the main objects of global optimization (see [8,19, 21,22] and references therein) In this subsection, as consequences of the results of Sect 6.1, we firstly derive Farkas lemma and stable Farkas lemma for systems involving convex composite functions and also DC functions Secondly, the strong duality for (P3), strong and stable strong duality results for the problem (P2) are given Farkas lemma and stable Farkas lemma for convex systems involving composite functions and DC functions As direct consequences of Theorem 16 and Corollary we get Theorem 18 The following statements are equivalent: (i) The following set is w ∗ -closed regarding the set X ∗ × {0U ∗ } × {0V ∗ } × R: {(u ∗ , v ∗ , r ) | (v ∗ , r ) ∈ epi q ∗ } {0 X ∗ } × ∗ u ∗ ∈U+ (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) (Farkas lemma with composite convex-DC functions) ∀h ∈ Γ (X ), M(x) ∈ −U+ ⇒ q ◦ N (x) − h(x) ≥ ⇐⇒ ∀w ∗ ∈ dom h ∗ , ∃u ∗ ∈ U+∗ , ∃v ∗ ∈ V+∗ ∩ dom q ∗ ∗ u ∗ ◦ M + v ∗ ◦ N (w ∗ ) − q ∗ (v ∗ ) ≤ h ∗ (w ∗ ) (iii) (Stable Farkas lemmas involving composite functions) ∀x ∗ ∈ X ∗ , ∀γ ∈ R, M(x) ∈ −U+ ⇒ q ◦ N (x) − x ∗ , x ≥ γ ∃u ∗ ∈ U+∗ , ∃v ∗ ∈ V+∗ ∩ dom q ∗ , u ∗ ◦ M + v ∗ ◦ N − x ∗ − q ∗ (v ∗ ) ≥ γ on X 123 J Glob Optim Corollary 10 The following statements are equivalent: (i) The following set is w ∗ -closed regarding the set {0 X ∗ } × {0U ∗ } × {0V ∗ } × R: {(u ∗ , v ∗ , r ) | (v ∗ , r ) ∈ epi q ∗ } {0 X ∗ } × ∗ u ∗ ∈U+ (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) (Farkas lemma involving composite convex functions) ∀γ ∈ R, M(x) ∈ −U+ ⇒ (q ◦ N )(x) ≥ γ ∃ u ∗ ∈ U+∗ , ∃v ∗ ∈ V+∗ ∩ dom q ∗ : u ∗ ◦ M + v ∗ ◦ N − q ∗ (v ∗ ) ≥ γ on X Strong and stable strong duality for (P3) and (P2) The closedness condition in Theorem 18 also characterizes the strong duality for (P3) and stable strong duality for the Problem (P2) Theorem 19 The following assertions are equivalent: (i) The following set is w ∗ -closed regarding the set X ∗ × {0U ∗ } × {0V ∗ } × R: {(u ∗ , v ∗ , r ) | (v ∗ , r ) ∈ epi q ∗ } {0 X ∗ } × ∗ u ∗ ∈U+ (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) ∀h ∈ Γ (X ), strong duality holds for (P3), i.e., inf M(x)∈−U+ q ◦ N (x) − h(x) = max inf ∗ w∗ ∈dom h ∗ u ∗ ∈U+ v ∗ ∈V+∗ ∩dom q ∗ h ∗ (w ∗ ) + q ∗ (v ∗ ) − u ∗ ◦ M ∗ +v ∗ ◦ N (w ∗ ) (iii) Stable strong duality for (P2) holds, i.e., for all x ∗ ∈ X ∗ , inf M(x)∈−U+ q ◦ N (x) − x ∗ = = max inf u ∗ ◦ M + v ∗ ◦ N − x ∗ − q ∗ (v ∗ ) , max − (u ∗ ◦ M + v ∗ ◦ N )∗ (x ∗ ) + q ∗ (v ∗ ) ∗ X u ∗ ∈U+ v ∗ ∈V+∗ ∩dom q ∗ ∗ u ∗ ∈U+ v ∗ ∈V+∗ ∩dom q ∗ Proof The conclusion is a direct consequences of Theorem 18 Strong duality result for (P2) now is straightforward from Corollary 10 Corollary 11 Assume that γ := inf(P2) ∈ R The following assertions are equivalent: 123 J Glob Optim (i) The following set is w ∗ -closed regarding the set {0 X ∗ } × {0U ∗ } × {0V ∗ } × R: {(u ∗ , v ∗ , r ) | (v ∗ , r ) ∈ epi q ∗ } {0 X ∗ } × ∗ u ∗ ∈U+ (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ , + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + (ii) Strong duality for (P2) holds, i.e., inf M(x)∈−U+ q ◦ N (x) = max inf u ∗ ◦ M + v ∗ ◦ N − q ∗ (v ∗ ) ∗ X u ∗ ∈U+ v ∗ ∈V+∗ ∩dom q ∗ Optimality condition for (P2) can be derived from the previous Farkas-type results as shown in the next theorem Theorem 20 Let x0 be a feasible point of (P2) Assume that the set {(u ∗ , v ∗ , r ) | (v ∗ , r ) ∈ epi q ∗ } {0 X ∗ } × ∗ u ∗ ∈U+ (x ∗ , −u ∗ , −v ∗ , r ) | (x ∗ , r ) ∈ epi (u ∗ ◦ M + v ∗ ◦ N )∗ + ∗ ×V ∗ (u ∗ ,v ∗ )∈U+ + is w ∗ -closed regarding the set {0 X ∗ } × {0U ∗ } × {0V ∗ } × R Then x0 is a solution of (P2) if and only if there exist v ∗ ∈ V+∗ ∩ dom q ∗ , u ∗ ∈ U+∗ such that v ∗ ∈ ∂q N (x0 ) , u ∗ ◦ M(x0 ) = 0, and ∈ ∂ u ∗ ◦ M + v ∗ ◦ N (x0 ) Proof • Necessary condition Assume that x0 is a solution of (P2) This is the same as M(x) ∈ −U+ ⇒ (q ◦ N )(x) ≥ (q ◦ N )(x0 ) (15) It now follows from the Farkas-type result just established in this section, namely Corollary 10, we get the existence of v ∗ ∈ V+∗ ∩ dom q ∗ and u ∗ ∈ U+∗ satisfying u ∗ ◦ M(x) + v ∗ ◦ N (x) − q ∗ (v ∗ ) ≥ q ◦ N (x0 ), ∀x ∈ X Since M(x0 ) ∈ −U+ , one gets holds at x0 , we get u∗ ◦ M(x0 ) ≤ Combining this and the fact that (16) u ∗ ◦ M(x0 ) = and v ∗ ◦ N (x0 ) − q ◦ N (x0 ) = q ∗ (v ∗ ) The last equality in ∗ (16) (17) means that v ∗ (17) ∈ ∂q N (x0 ) It now follows from (16) and (17), ∗ u ◦ M(x) + v ◦ N (x) ≥ u ∗ ◦ M(x0 ) + v ∗ ◦ N (x0 ), ∀x ∈ X, which shows that ∈ ∂ u ∗ ◦ M + v ∗ ◦ N (x0 ) • sufficient condition Assume that there are v ∗ ∈ V+∗ ∩ dom q ∗ and u ∗ ∈ U+∗ satisfying the conditions of the theorem Then it is easy to see that (16) holds and again, Corollary 10 ensures that (15) holds, showing that x0 is a solution of (P2) Remark We can also include a geometric constraint of the form x ∈ C, where C is a closed convex subset of X , to the Problems (P1), (P2), and (P3) by taking g = i C The same 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